The defining aspect of NBG is the distinction between proper class and set. Let a and s be two individuals. Then the atomic sentence is defined if a is a set and s is a class. In other words, is defined unless a is a proper class. A proper class is very large; NBG even admits of "the class of all sets", the universal class called V. However, NBG does not admit "the class of all classes" (which fails because proper classes are not "objects" that can be put into classes in NBG) or "the set of all sets" (whose existence cannot be justified with NBG axioms).

By NBG's axiom schema of Class Comprehension, all objects satisfying any given formula in the first-order language of NBG form a class; if a class is not a set in ZFC, it is an NBG proper class.

The development of classes mirrors the development of naive set theory. The principle of abstraction is given, and thus classes can be formed out of all individuals satisfying any statement of first-order logic whose atomic sentences all involve either the membership relation or predicates definable from membership. Equality, pairing, subclass, and such, are all definable and so need not be axiomatized — their definitions denote a particular abstraction of a formula.

Sets are developed in a manner very similarly to ZF. Let Rp(A,a), meaning "the set a represents the class A," denote a binary relation defined as follows:

That is, a "represents" A if every element of a is an element of A, and conversely. Classes lacking representations, such as the class of all sets that do not contain themselves (the class invoked by the Russell paradox), are the proper classes.

In two articles published in 1925 and 1928, John von Neumann stated his axioms and showed they were adequate to develop set theory.[1] Von Neumann took functions and arguments as primitives. His functions correspond to classes, and functions that can be used as arguments correspond to sets. In fact, he defined classes and sets using functions that can take only two values (that is, indicator functions whose domain is the class of all arguments).

Von Neumann's work in set theory was influenced by Cantor's articles, Zermelo's 1908 axioms for set theory, and the 1922 critiques of Zermelo's set theory that were given independently by Fraenkel and Skolem. Both Fraenkel and Skolem pointed out that Zermelo's axioms cannot prove the existence of the set {Z0, Z1, Z2, … } where Z0 is the set of natural numbers, and Zn+1 is the power set of Zn. They then introduced the axiom of replacement, which would guarantee the existence of such sets.[2] However, they were reluctant to adopt this axiom: Fraenkel's opinion was "that Replacement was too strong an axiom for 'general set theory' … and … Skolem only wrote that 'we could introduce' Replacement".[3]

Von Neumann worked on the deficiencies in Zermelo's set theory and introduced several innovations to remedy them, including:

A theory of ordinals. Zermelo's set theory does not contain Cantor's theory of ordinal numbers. Von Neumann recovered this theory by defining the ordinals using sets that are well-ordered by the ∈-relation.[4] In contrast to Fraenkel and Skolem, von Neumann found the axiom of replacement so essential to his work that he declared: "In fact, I believe that no theory of ordinals is possible at all without this axiom."[5]

A criterion identifying classes that are too large to be sets. Zermelo did not provide such a criterion. His set theory avoids the large classes that lead to the paradoxes, but it leaves out many sets, such as the one mentioned by Fraenkel and Skolem.[6] Von Neumann's criterion is: A class is too large to be a set if and only if it can be mapped onto the universal class. Von Neumann realized that the paradoxes can be avoided by not allowing such large classes to be members of any class. Combining this restriction with his criterion, he obtained his axiom of limitation of size: A class X is not a member of any class if and only if X can be mapped onto the universal class.[7] He proved that this axiom implies the axioms of replacement and separation, and implies that the universal class can be well-ordered (which is equivalent to the axiom of global choice).

Finite axiomatization. Fraenkel and Skolem formalized Zermelo's imprecise concept of "definite propositional function", which appears in his axiom of separation. Skolem gave the axiom schema of separation that is currently used in ZFC; Fraenkel gave an equivalent approach. Zermelo rejected both approaches "particularly because they implicitly involve the concept of natural number which, in Zermelo's view, should be based upon set theory."[8] Von Neumann avoided axiom schemas by formalizing the concept of "definite propositional function" with his functions, whose construction requires only finitely many axioms. This led to his set theory having finitely many axioms.[9] (In 1961, Montague proved that ZFC cannot be finitely axiomatized.[10])

The axiom of regularity. Zermelo's set theory does not exclude non-well-founded sets.[11] Fraenkel and von Neumann introduced axioms to exclude these sets. Von Neumann introduced the axiom of regularity, which states that all sets are well-founded.[12] Although von Neumann did not adopt regularity as an axiom,[13] he proved its relative consistency while studying his axiom of limitation of size. First he weakened his axiom system by replacing the latter axiom with two of its consequences: replacement and a choice axiom equivalent to global choice.[14] Next he proved that if this weaker system is consistent, it remains consistent after adding the axiom of regularity. Finally, he showed that his weaker system augmented with regularity proves the axiom of limitation of size.[15] These results establish that the axioms of regularity and limitation of size are relatively consistent with respect to his weaker system, and that (in the presence of regularity and his other axioms) replacement and his choice axiom are equivalent to the axiom of limitation of size.

In a series of articles published between 1937 and 1954, Paul Bernays modified von Neumann's theory by taking sets and classes as primitives. By using sets, Bernays was following the tradition established by Cantor, Dedekind, and Zermelo. His classes followed the tradition of Boolean algebra since they permit the operation of complement as well as union and intersection.[16] Bernays handled sets and classes in a two-sorted logic. This required the introduction of two membership primitives: one for membership in sets, and one for membership in classes. With these primitives, Bernays rewrote and simplified von Neumann's axioms. He also adopted the axiom of regularity, and replaced the axiom of limitation of size with the axioms of replacement and von Neumann's choice axiom. (Von Neumann's work shows that the last two changes allow Bernays' axioms to prove the axiom of limitation of size.) [17]

Kurt Gödel simplified Bernays' theory by making every set a class, which allowed him to use just one sort for classes and one membership primitive. He also introduced a predicate indicating which classes are sets. Gödel modified some of Bernays' axioms, and introduced the axiom of global choice to replace von Neumann's choice axiom. He used his axioms in his 1940 monograph on the relative consistency of global choice and the generalized continuum hypothesis.[18]

Several reasons have been given for Gödel choosing NBG for his 1940 monograph.[19] Gödel gave a mathematical reason—NBG's global choice produces a stronger consistency theorem: "This stronger form of the axiom [of choice], if consistent with the other axioms, implies, of course, that a weaker form is also consistent."[20]Robert Solovay conjectured: "My guess is that he wished to avoid a discussion of the technicalities involved in developing the rudiments of model theory within axiomatic set theory."[21]Kenneth Kunen gave a reason for Gödel avoiding this discussion: "There is also a much more combinatorial approach to L [the constructible universe], developed by … [Gödel in his 1940 monograph] in an attempt to explain his work to non-logicians. … This approach has the merit of removing all vestiges of logic from the treatment of L."[22]Charles Parsons gives a philosophical reason for Gödel's choice of NBG: "This view [that 'property of set' is a primitive of set theory] may be reflected in Gödel's choice of a theory with class variables as the framework for … [his monograph]."[23]

Gödel's achievement together with the details of his presentation led to the prominence that NBG would enjoy for the next two decades.[24] Even Paul Cohen's 1963 independence proofs for ZF used tools that Gödel developed for his work in NBG.[25] However, in the 1960s, ZFC became more popular than NBG. This was caused by several factors, including the extra work required to handle forcing in NBG,[26] Cohen's 1966 presentation of forcing (which uses techniques that naturally belong to ZF),[27] and the proof that NBG is a conservative extension of ZFC.[28]

NBG is presented here as a two-sorted theory, with lower case letters denoting variables ranging over sets, and upper case letters denoting variables ranging over classes. Hence "" should be read "set x is a member of set y," and "" as "set x is a member of class Y." Statements of equality may take the form "" or "". "" stands for "" and is an abuse of notation. NBG can also be presented as a one-sorted theory of classes, with sets being those classes that are members of at least one other class.

We first axiomatize NBG using the axiom schema of Class Comprehension. This schema is provably equivalent[29] to 9 of its finite instances, stated in the following section. Hence these 9 finite axioms can replace Class Comprehension. This is the precise sense in which NBG can be finitely axiomatized.

pairing: For any sets x and y, there is a set, , whose elements are exactly x and y.

pairing implies that for any set x, the set {x} (the singleton set) exists. Also, given any two sets x and y and the usual set-theoretic definition of the ordered pair, the ordered pair (x,y) exists and is a set. By Class Comprehension, all relations on sets are classes. Moreover, certain kinds of class relations are one or more of functions, injections, and bijections from one class to another. pairing is an axiom in Zermelo set theory and a theorem in ZFC.

union: For any set x, there is a set which contains exactly the elements of elements of x.

power set: For any set x, there is a set which contains exactly the subsets of x.

infinity: There exists an inductive set, namely a set x whose members are (i) the empty set; (ii) for every member y of x, is also a member of x.

infinity can be formulated so as to imply the existence of the empty set.[30]

The remaining axioms have capitalized names because they are primarily concerned with classes rather than sets. The next two axioms differ from their ZFC counterparts only in that their quantified variables range over classes, not sets:

Extensionality:: Classes with the same elements are the same class.

Foundation (Regularity): Each nonempty class is disjoint from one of its elements.

The last two axioms are peculiar to NBG:

Limitation of Size: For any class C, a set x such that x=C exists if and only if there is no bijection between C and the class V of all sets.

From this axiom, due to Von Neumann, Subsets, Replacement, and Global Choice can all be derived. This axiom implies the axiom of global choice because the class of ordinals is not a set; hence there exists a bijection between the ordinals and the universe. If Limitation of Size were weakened to "If the domain of a class function is a set, then the range of that function is likewise a set," then no form of the axiom of choice is an NBG theorem. In this case, any of the usual local forms of Choice may be taken as an added axiom, if desired.

Class Comprehension is the only axiom schema of NBG. In the next section, we show how this schema can be replaced by a number of its own instances. Hence NBG can be finitely axiomatized. If the quantified variables in φ(x) range over classes instead of sets, the result is Morse–Kelley set theory, a proper extension of ZFC which cannot be finitely axiomatized.

An appealing but somewhat mysterious feature of NBG is that its axiom schema of Class Comprehension is equivalent to the conjunction of a finite number of its instances. The axioms of this section may replace the Axiom Schema of Class Comprehension in the preceding section. The finite axiomatization presented below does not necessarily resemble exactly any NBG axiomatization in print.

We develop our axiomatization by considering the structure of formulas.

Sets: For any set x, there is a class X such that x=X.

This axiom, in combination with the set existence axioms from the previous axiomatization, assures the existence of classes from the outset, and enables formulas with class parameters.

We now turn to quantification. In order to handle multiple variables, we need the ability to represent relations. Define the ordered pair as as usual. Note that three applications of pairing to a and b assure that (a,b) is indeed a set.

Products: For any classes A and B, the class is a class. (In practice, only is needed.)

Converses: For any class R, the classes:

and

exist.

Association: For any class R, the classes:

and

exist.

These axioms license adding dummy arguments, and rearranging the order of arguments, in relations of any arity. The peculiar form of Association is designed exactly to make it possible to bring any term in a list of arguments to the front (with the help of Converses). We represent the argument list as (it is a pair with the first argument as its first projection and the "tail" of the argument list as the second projection). The idea is to apply Assoc1 until the argument to be brought to the front is second, then apply Conv1 or Conv2 as appropriate to bring the second argument to the front, then apply Assoc2 until the effects of the original applications of Assoc1 (which are now behind the moved argument) are corrected.

Mendelson[32] refers to his axioms B1-B7 of class comprehension as "axioms of class existence." Four of these identical to axioms already stated above: B1 is Membership; B2, Intersection; B3, Complement; B5, Product. B4 is Ranges modified to assert the existence of the domain of R (by existentially quantifying y instead of x). The last two axioms are:

B6:

B7:

B6 and B7 enable what Converses and Association enable: given any class X of ordered triples, there exists another class Y whose members are the members of X each reordered in the same way.

For a discussion of some ontological and other philosophical issues posed by NBG, especially when contrasted with ZFC and MK, see Appendix C of Potter (2004).

Even though NBG is a conservative extension of ZFC, a theorem may have a shorter and more elegant proof in NBG than in ZFC (or vice versa). For a survey of known results of this nature, see Pudlak (1998).

The ontology of NBG provides scaffolding for speaking about "large objects" without risking paradox. In some developments of category theory, for instance, a "large category" is defined as one whose objects make up a proper class, with the same being true of its morphisms. A "small category", on the other hand, is one whose objects and morphisms are members of some set. We can thus easily speak of the "category of all sets" or "category of all small categories" without risking paradox. Those categories are large, of course. There is no "category of all categories" since it would have to contain large categories which no category can do. Although yet another ontological extension can enable one to talk formally about such a "category" (see for example the "quasicategory of all categories" of Adámek et al. (1990), whose objects and morphisms form a "proper conglomerate").

On whether an ontology including classes as well as sets is adequate for category theory, see Muller (2001).

^von Neumann 1923. Von Neumann's definition also used the theory of well-ordered sets. Later, his definition was simplified to the current one: An ordinal α is a set that is well-ordered by ∈ and has the property that every member of α is a subset of α (Kunen 1980, p. 16).

^Von Neumann also analyzed Fraenkel's axiom and stated that it was not "precisely formulated", but it would approximately say: "Besides the sets … whose existence is absolutely required by the axioms, there are no further sets." His analysis led him to reject this axiom and to propose the axiom of regularity instead (von Neumann 1925, pp. 230–232,239; English translation: p. 404–405 & p. 410–411).

^Gödel used von Neumann's axioms in his 1938 announcement of his relative consistency theorem, and used ZF in his 1939 outline of his proof (Ferreirós 2007, p. 382). In his 1938 announcement, Gödel also stated "A corresponding theorem holds if T denotes the system Principia mathematica …" (Gödel 1990, p. 26). Proving a theorem in multiple formal systems was not unusual for Gödel. For example, he proved his incompleteness theorem for the system of Principia mathematica, but pointed out that it "holds for a wide class of formal systems …" (Gödel 1986, p. 145).

^Gödel 1990, p. 13. Gödel's consistency proof builds the constructible universe. To build this in ZF requires some model theory; Gödel built it in NBG without model theory. For a discussion of Gödel's technique: see Cohen 1966, pp. 99–103.

^Gödel 1990, p. 108, footnote i. The paragraph containing this footnote discusses why Gödel considered "property of set" a primitive of set theory and how it fit into his ontology. "Property of set" corresponds to the "class" primitive in NBG.

^Kanamori 2009, p. 23: "Forcing itself went a considerable distance in downgrading any formal theory of classes because of the added encumbrance of having to specify the classes of generic extensions."